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If two instrument variables (for example $Z_1$ and $Z_2$) are both exogenous and relevant. And we found Wu-Hausman test fails to reject the $H_0$: Both IV regression and OLS are consistent. When we want to report the OLS result:

We have two options: $$(1) Y = X_1 + X_2 + ... + Z_1 + Z_2 + e $$ $$(2) Y = X_1 + X_2 + ... + e $$ Is there any compulsory reasons that $Z_1 + Z_2$ need to be in the equation? Thanks very much!

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The null hypothesis of the Wu-Hausman test is that both estimators are consistent. When you use statistical tests, you either reject the null, or you fail to reject the null, but you cannot accept the null. Therefore, you cannot use the result of this test to support the claim that the OLS is consistent.

An example for this is that, if you use weak instruments and the precision of your IV estimator is poor, you will frequently end up with a Wu-Hausman test that fails to reject the null hypothesis. This happens not because the point estimates are close, but because the standard error of the IV estimator is large.

Look at both the OLS and the IV estimates, and in particular, look at their precisions. If they are imprecise, that might explain why the test does not reject the null. If both are very precise (and close to each other), then you have no reason not to report it, because it is an interesting result. In general, my advice would be to report both the OLS (without including $Z_1$ and $Z_2$ among the regressors) and the IV estimate results (as well as the result of the Wu-Hausman test).

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  • $\begingroup$ Thanks very much, Roland for the answer. To be more rigorous, if the Wu-Hausman test fails to reject the null that the OLS is consistent, should we still need to include the instrument variable, which is a strong instrument, as a control variable in the OLS? $\endgroup$
    – Ray Yang
    Sep 13, 2018 at 0:48
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    $\begingroup$ The WH test works under the assumption that the IV estimator is consistent, therefore it is equivalent to say that it tests that the OLS is consistent or that both are consistent. As I state above, you may fail to reject the null because of a lack of power, in which case point estimates of OLS and IV do not need to be very close. My advice (as above): no, don't get rid of the IV estimate, show both. $\endgroup$
    – Roland
    Sep 13, 2018 at 7:55
  • $\begingroup$ Hi Roland, Thanks for the clarification. As I've re-edited the question, should we include the IVs as control variables in the OLS when we want to report the OLS result? $\endgroup$
    – Ray Yang
    Sep 13, 2018 at 17:04
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    $\begingroup$ Hi Ray, Thanks for clarifying the question, I understand better now. The usual practice would be not to include $Z_1$ and $Z_2$ in the OLS regression. I'll edit the answer to make this clear. $\endgroup$
    – Roland
    Sep 13, 2018 at 22:21
  • $\begingroup$ @Roland, what if the instrument(s) is(are) not weak? Maybe there is endogeneity, but it is so small that the OLS is fine, that is theoretically possible, or? $\endgroup$
    – Fuca26
    Apr 9, 2021 at 16:53

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